User blog:B1mb0w/new Alpha Function
'REPLACED' This blog has been replaced. A new Alpha Function blog should be referred to instead. The blog can be found here. '(previous) Alpha Function' The Alpha Function has one parameter: \(\alpha®\) where r is any real number. It is derived from the Strong D Function with a variable number of input parameters. This blog replaces the old description of the Alpha Function. 'What is the Alpha Function' My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(1)\) for example can be used to reference the number 2. Therefore 1 is the Alpha Index for the number 2. Alpha needs to reference big numbers very quickly to be useful, therefore it uses the Strong D Function for this purpose. Alpha should also be strictly hierarchical and every number a > b, must reference larger numbers, so that \(\alpha(a) >> \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and offers some finesse to locate large big numbers. The Alpha Function is defined recursively based on a series of binary decisions. The logic can be explained by referring to the following notation. 'Some Notation' Following notation helps to explain the behaviour of Strong D Functions and the logic of the Alpha Function. \(D(m_{x}) = D(m_1,m_2,...,m_x)\) \(D(m_{x},n_{y}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\) \(D(1,0_{y}) = D(D(1_{y})_{y})\) \(D(m,n_{y}) = D(m-1,D(m,n_{y-1},n_y-1)_{y})\) \(D(m_{x},n,0_{y}) = D(m_1-1,D(m_{x},n-1,n_{y})_{x+y})\) when x>0 and which is equal to \(= D(m_{x},n-1,n_{y-1},n_y+1)\) 'Alpha Function Logic' The Alpha Function is defined using the following logic. \(\alpha® = D(D(m_{x})_{x})\) where \(2^{x-1} <= r < 2^x\) and \(\alpha(2^x) = D(1, 0_{x})\) The values of \(m_{x}\) are calculated based on the value of r but only legal values can be used which follow these restrictions: *duplicate values should be avoided *out of sequence results must be avoided The second constraint is important to force \(\alpha(a) >> \alpha(b)\) whenever a > b. Additional logic is derived from the following rules: Maximum Value Rule: M1 \(D(1,0_{x}) = D(D(1_{x})_{x})\) therefore \(D(m_{x}) < D(D(1_{x})_{x})\) \(<= D(D(1_{x})_{x})-1\) or alternatively \(<= D(D(1_{x})_{x-1},D(1_{x-1},0))\) Rule M1a and \(m_1 >= 1\) Rule M1b Maximum Value Rule: M2 \(D(m_{x},n+1,0_{y}) = D(m_{x},n,n_{y-1}+1,n_y+2)\) therefore \(D(m_{x},n,p_{y}) < D(m_{x},n,n_{y-1}+1,n_y+2)\) \(<= D(m_{x},n,n_{y-1}+1,n_y+2)-1\) or alternatively \(<= D(m_{x},n,n_{y}+1)\) Rule M2a or \(p_{y} <= n_{y}+1\) Rule M2b Maximum Value Rule: M3 It can also be shown that the only legal values for D functions in the form: \(D(m_{x})\) are when \(m_i <= m_{i-1}+1\) for all \(3 <= i <= x\) Maximum Value Rule: M4 The final rule is used for D functions of the form: \(D(n+1,0_{y}) = D(n,D(n,n_{y}+1)_{y}\) therefore \(D(n,p_{y}) < D(n,D(n,n_{y}+1)_{y})\) \(<= D(n,D(n,n_{y}+1)_{y}-1)\) or alternatively \(<= D(n,D(n,n_{y}+1)_{y-1},D(n,n_{y-1}+1,n))\) Rule M4a ''' and \(p_i <= D(n,n_{y}+1)\) for all \(1 <= i < y\) '''Rule M4b and \(p_y <= D(n,n_{y-1}+1,n)\) Rule M4c 'Some Calculations' Program Code was used to calculate these examples. \(\alpha(0.00) = D() = 0\) \(\alpha(0.50) = D(0) = 1\) \(\alpha(1.00) = D(1) = 2\) \(\alpha(2.00) = D(1,0) = 3\) \(\alpha(2.08) = D(1,1) = 4\) \(\alpha(2.16) = D(1,2) = 5\) \(\alpha(2.56) = D(2,1) = 11\) \(\alpha(e) = D(2,6) = 26\) \(\alpha(3.00) = D(3,0) = 59\) \(\alpha(3.04) = D(3,2) = 563\) \(\alpha(\pi) = D(3,14) = 301327043\) \(\alpha(3.52) = D(4,0) >>\) Googol \(\alpha(3.6) = D(4,1) >> f_{\omega}(3)\) \(\alpha(3.64) = D(4,2) >>\) Googolplex \(\alpha(4.00) = D(1,0,0)\) \(\alpha(4.0005) = D(1,0,1) >> g_2\) where \(g_{64}\) is Graham's number \(\alpha(4.008304) = D(1,11,0) >> D(1,9,9) >>\) Graham's number \(\alpha(4.1250) = D(2,0,0) >> f_{\omega+1}^2(3)\) \(\alpha(4.1255) = D(2,0,1) >> f_{\omega+2}(3)\) \(\alpha(4.2501) = D(3,0,1) >> f_{\omega.2}(3)\) \(\alpha(6) = D(D(1,0,0),0,0) = D(\alpha(4),0,0)\) \(\alpha(6.5) = D(D(1,0,1),0,0) = D(D(D(4,4),D(4,4)),0,0)\) \(\alpha(8) = D(1,0,0,0) = D(D(1,1,1),D(1,1,1),D(1,1,1))\) \(\alpha(8.tba) = D(D(3,0,1),0,0) >> f_{\omega.2+1}(3)\) This result is being checked. Work in progress \(\alpha(9) = D(D(59,0,0),0,0,0) = D(\alpha(5),0,0,0)\) \(\alpha(16) = D(1,0,0,0,0)\) \(\alpha(32) = D(1,0,0,0,0,0)\) \(\alpha(48) = D(D(1,0,0,0,0,0),0,0,0,0,0) = D(\alpha(32),0,0,0,0,0)\) \(\alpha(125) = D(D(1,1,1,1,0,1,0),0,0,0,0,0,0)\) \(\alpha(128) = D(1,0,0,0,0,0,0,0)\) 'Program Code and Description' Version 1 of the program code for the Alpha Function is available here. The code is not complete and various errors will be corrected in Version 2 (work in progress). 'Comments and Questions' Look forward to comments and questions. I am learning heaps by writing these blogs and correcting all the mistakes the community finds in them ! Cheers B1mb0w. 'References' The Alpha Function *Detailed Alpha Function Calculations Thanks to Nayuta Ito *Strong D Function **Deeply Nested Ackermann Function ***Modified Ackermann Function **Summary of Strong D Function Growth Rates ***General Proof of Strong D Function Growth Rate **Strong D Function and f epsilon nought (n) *Alpha Function and f epsilon nought *Alpha Function Program Code **Version 2 Work in Progress **Version 1 *''The (old) Alpha Function'' **Detailed Alpha Function Calculations Thanks to Nayuta Ito Category:Blog posts